[Paper Review] Pointwise Stabilization of Discrete-time Stationary Matrix-valued Markovian Chains

Xiongping Dai, Yu Huang+1 more|arXiv (Cornell University)|Jul 1, 2011

Stability and Control of Uncertain Systems24 references3 citations

TL;DR

This paper establishes that for discrete-time, stationary matrix-valued Markovian jump linear systems, pointwise convergence and pointwise exponential convergence are equivalent under the condition of essential nonuniform product boundedness. By leveraging measure theory, ergodic theory, and a state space splitting theorem, the authors prove that almost sure convergence to zero for each initial state implies exponential convergence rates, providing a key characterization in symbolic dynamics and stochastic stability theory.

ABSTRACT

We study the pointwise stabilizability of a discrete-time, time-homogeneous, and stationary Markovian jump linear system. By using measure theory, ergodic theory and a splitting theorem of state space we show in a relatively simple way that if the system is essentially product-bounded, then it is pointwise convergent if and only if it is pointwise exponentially convergent.

Motivation & Objective

• To characterize the conditions under which pointwise convergence implies pointwise exponential convergence in discrete-time Markovian jump linear systems.
• To address the non-equivalence of pointwise and exponential convergence in stochastic systems, a long-standing challenge in control theory.
• To establish a theoretical framework using symbolic dynamics and ergodic theory to analyze convergence behavior under probabilistic switching.
• To clarify the distinction between uniform and non-uniform convergence in linear cocycles and Markovian systems.
• To provide a necessary and sufficient condition—essentially nonuniform product boundedness—under which pointwise convergence implies exponential convergence.

Proposed method

• Utilizes a splitting theorem of the state space ℝ¹ˣᵈ into stable and unstable subspaces based on Lyapunov exponents.
• Applies measure-theoretic tools and the ergodic decomposition theorem to analyze invariant measures on symbolic sequences.
• Employs the shift map θ on the one-sided shift space Σ⁺ₖ to model trajectory evolution and analyze long-term behavior.
• Introduces the notion of essential nonuniform product boundedness to control growth of matrix products across switching sequences.
• Leverages Egoroff’s theorem to derive almost uniform convergence on subsets of positive measure, enabling exponential rate estimates.
• Uses the stable subspace Es(i·) associated with each trajectory i· to characterize convergence behavior via spectral properties.

Experimental results

Research questions

• RQ1 Under what conditions does pointwise convergence imply pointwise exponential convergence in discrete-time Markovian jump linear systems?
• RQ2 How does the structure of the Markovian switching process affect the stability of matrix products?
• RQ3 What is the role of ergodicity and invariant measures in characterizing convergence types in stochastic linear systems?
• RQ4 Can non-uniform convergence be transformed into exponential convergence under mild boundedness assumptions?
• RQ5 Why is the distinction between uniform and non-uniform convergence critical in linear cocycle theory and control systems?

Key findings

• Pointwise convergence and pointwise exponential convergence are equivalent if the system is essentially nonuniformly product bounded.
• The existence of a positive-measure set of trajectories for which xSᵢ₁⋯Sᵢₙ → 0 implies the existence of a subset where exponential decay occurs.
• For any nonzero initial state x, there exists a Borel subset of trajectories with positive measure such that x lies in the stable subspace Es(i·).
• The stable subspace Es(i·) is invariant under the shift map θ, and convergence on this subspace ensures exponential decay rates.
• Even under ergodicity, the set of trajectories achieving pointwise convergence may not have full measure, highlighting the non-uniform nature of convergence.
• The proof relies on the interplay between ergodic decomposition, almost uniform convergence (via Egoroff’s theorem), and spectral splitting of the state space.